Finite Group Databasebeta

← #10692449

Subgroups — #10692449

Order 768.

418 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83217C211yesmaximal, characteristic
2#79016C211yesmaximal, characteristic
3C3×#1988C211yesmaximal, characteristic, complemented
4#71777C211yesmaximal, characteristic, complemented
5#69470C211yesmaximal, characteristic
6#67708C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9375—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5479C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5111—21no—
14C8×#67—21no—
15#5091—21no—
16#5089C412yescomplemented
17#5084—21no—
18#5081C412yescomplemented
19#4971—21no—
20#4967—21no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4840—21no—
25#4839—21no—
26#4836—21no—
27#4828—21no—
28C3×#340—21no—
29C3×#340—21no—
30#4546—21no—
31#4546—21no—
32#4452—21no—
33#4451—21no—
34#3238—31no—
35#2992—31no—
36#1988D311yescharacteristic, complemented
37#1558—31no—
38#1509—31no—
39#1290—31no—
40#1285—31no—
41C22×C3×Q8—21no—
42C22×C3×C8D411yescharacteristic
43C22×C3×C8D411yescharacteristic
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8C2×C412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53#930D411yescharacteristic
54#929—21no—
55C4×#67—21no—
56#924D411yescharacteristic
57C2×ΓL(1,𝔽25)—21no—
58C2×D3×C8—21no—
59#876—21no—
60#875C2×C411yescharacteristic
61C4×D12D411yescharacteristic
62#872—21no—
63#870C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×#95—21no—
68#830—21no—
69#830—21no—
70#830—21no—
71(C2×D3)⋊C8—21no—
72(C2×D3)⋊C8—21no—
73#815—21no—
74Dic3⋊C8—21no—
75Dic3⋊C8—21no—
76C8×Dic3—21no—
77#810—22no—
78#809—22no—
79#808—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D411yescharacteristic
84C4⋊(C3⋊C8)D411yescharacteristic
85#803—21no—
86#803—21no—
87#803C2×C412yes—
88C2×C4⋊Q8C2×D311yescharacteristic
89#529—31no—
90#501—32no—
91#489—32no—
92#483—61no—
93#480—61no—
94#477—61no—
95#476—61no—
96#446—61no—
97#435—61no—
98#435—61no—
99C8×D4—61no—
100#433—61no—
101#432—31no—
102#432—31no—
103#432—31no—
104#424—61no—
105C2×C4⋊C8C2×D311yescharacteristic
106C4×C4⋊C4C2×D311yescharacteristic
107#340—21no—
108#340—21no—
109#333—61no—
110#333—61no—
111#328—61no—
112#327—61no—
113C2×C3×Q8—22no—
114C2×C3×Q8—41no—
115C2×C3×Q8—81no—
116C2×C3×Q8—81no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—21no—
123C2×D12C22⋊C411yescharacteristic
124C2×C4×D3—21no—
125C2×Dic6C22⋊C411yescharacteristic
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42—81no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141#217—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—21no—
146C4⋊Dic3C22⋊C411yescharacteristic
147C4⋊Dic3C22⋊C411yescharacteristic
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152C4×Dic3—21no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—41no—
157ΓL(1,𝔽25)—81no—
158D3×C8—81no—
159C22×Q8—21no—
160C2×M16—61no—
161C22×C8—61no—
162C22×C8#6711yescharacteristic
163C22×C8#6711yescharacteristic
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C4×D312yes—
168#126—61no—
169#125—31no—
170#124—31no—
171#123—61no—
172C22⋊Q8—31no—
173#121—31no—
174C4×Q8—31no—
175C4×D4—31no—
176C4×D4—61no—
177#117—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C42—21no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—31no—
188C4⋊C8—31no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—62no—
194#102—62no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200#97—32no—
201#97—61no—
202#97—61no—
203#97—61no—
204C4×C8—61no—
205#95—21no—
206C23×C3#12111yescharacteristic
207C22×D3—21no—
208C3×Q8—82no—
209C3×Q8—82no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4C2×C22⋊C411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12311yescharacteristic
227#67—41no—
228#67—41no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232D12—41no—
233C4×D3—41no—
234C4×D3—41no—
235Dic6—41no—
236C3×C8—41no—
237C3×C8—41no—
238C3⋊C8—42no—
239C3⋊C8—41no—
240C3⋊C8—41no—
241C3⋊C8—81no—
242C2×Q8—22no—
243C2×Q8—31no—
244C2×Q8—41no—
245C2×Q8—81no—
246C2×Q8—81no—
247C2×D4—31no—
248C2×D4—61no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—61no—
252C22×C4D3×D411yescharacteristic
253C22×C4#23711yescharacteristic
254C22×C4C2×#6711yescharacteristic
255M16—241no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—62no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—121no—
264C2×C8—241no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C42—61no—
283C42—81no—
284C42C2×C4×D311yescharacteristic
285C42#23511yescharacteristic
286C42D3×D411yescharacteristic
287C42D3×D411yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3#38611yescharacteristic
291C22×C3#47111yescharacteristic
292C22×C3#47211yescharacteristic
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—82no—
303C3×C4#45511yescharacteristic
304C3×C4#45611yescharacteristic
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308C23—61no—
309C23#93811yescharacteristic
310Q8—82no—
311Q8—82no—
312Q8—121no—
313D4—121no—
314D4—121no—
315D4—121no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4D3×C22⋊C411yescharacteristic
333C2×C4#67⋊C411yescharacteristic
334C2×C4#88311yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4C4×#6711yescharacteristic
337C2×C4#93711yescharacteristic
338C8—41no—
339C8—41no—
340C8—122no—
341C8—121no—
342C8—121no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#188211yescharacteristic
347C2×C3#190711yescharacteristic
348C2×C3#190911yescharacteristic
349D3—81no—
350C22—21no—
351C22—21no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22#495611yescharacteristic
356C22#516011yescharacteristic
357C22#517711yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—82no—
365C4—121no—
366C4—121no—
367C4—121no—
368C4#506111yescharacteristic
369C4#509511yescharacteristic
370C3#937511yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—241no—
374C2#6969311yescharacteristic
375C2#7021811yescharacteristic
376C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1988D3
hypocenterC3#9375
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1988

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C3×#1988C2semidirect
4. #71777C2semidirect
6. #67708C2semidirect
7. #67621C2semidirect
16. #5089C4semidirect
18. #5081C4semidirect
36. #1988D3semidirect
370. C3#9375semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#832172
#790162
C3×#19882
#717772
#694702
#677082
#676212
#93753